U14 Linear Equations in Two Unknowns: Essential Formulas

A system of linear equations in two unknowns involves finding the values of $x$ and $y$ that satisfy two equations simultaneously. The solution represents the point of intersection of two straight lines on the Cartesian plane.

1 General Form and Solution Methods

General Form

A system of two linear equations in two unknowns $x$ and $y$ is typically written as:

$$ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases} $$

where $a_1$, $b_1$, $c_1$, $a_2$, $b_2$, $c_2$ are real numbers, and $a_1$ and $b_1$ are not both zero, similarly for $a_2$ and $b_2$.

Graphical Interpretation

Each equation represents a straight line on the $xy$-plane. The solution $(x, y)$ to the system corresponds to the point of intersection of these two lines.

(x, y) $a_1x+b_1y=c_1$ $a_2x+b_2y=c_2$

2 Algebraic Solution Methods

Method of Substitution

Solve one equation for one variable (e.g., $y$ in terms of $x$), then substitute this expression into the other equation to find the value of the first variable.

$$ \text{If } y = mx + d \text{ from Eqn (1), substitute into Eqn (2): } a_2x + b_2(mx + d) = c_2 $$

Method of Elimination

Multiply the equations by suitable constants so that the coefficients of one variable become equal in magnitude but opposite in sign. Adding the equations then eliminates that variable.

$$ \begin{aligned} &k \times (a_1x + b_1y = c_1) \\ &l \times (a_2x + b_2y = c_2) \\ &\text{such that } k b_1 + l b_2 = 0 \text{ (to eliminate $y$).} \end{aligned} $$

3 Nature of Solutions

Unique Solution (Intersecting Lines)

The lines intersect at exactly one point. This occurs when the ratios of the coefficients are not equal.

$$ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} $$

No Solution (Parallel Lines)

The lines are parallel and never intersect. This occurs when the ratios of the $x$ and $y$ coefficients are equal, but not equal to the ratio of the constants.

$$ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} $$

Infinitely Many Solutions (Coincident Lines)

The two equations represent the same line. Every point on the line is a solution. This occurs when all three ratios are equal.

$$ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} $$

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